Relative homological algebra has been formulated by Hochschild in
categories of modules and later by Heller, Butler and Horrocks in more general categories with a relative abelian structure. Beligiannis developed in [1]} a relative version of homological algebra in triangulated categories in analogy to relative homological algebra in abelian categories, in which the notion of a proper class of exact sequences is replaced by a proper class of triangles. By specifying a class of triangles \(\xi \) , which is called a proper class of triangles, he introduced the full subcategory \(\mathcal {P}(\xi )\)  of \(\xi \) -projective objects and the full subcategory \(\mathcal {I}(\xi )\)  of \(\xi \) -injective objects
